STAT331 Final Project Report

Author

Thien An Tran, Tejasree Kandibanda, Matthew Huang, Chloe Anbarcioglu

TODO: - update interpretations to log of average murder rate - add region colors to plots - conclusion

1 Introduction

TODO: - add more to intro

For our project, we want to explore the relationship between murder rate and happiness score in each country per year

We obtained our data from Gapminder (n.d.).

population: contains total population counts for each country per year.
murder: contains total number of estimated deaths from interpersonal violence for each country per year.
happiness: contains happiness score (converted to 0 to 100 scale to be in terms of percentage) for each country per year.

TODO: - Variables selected (define explanatory + response) - Explain why variables selected

Using these datasets, the variables of interest that we will use to answer our questions are:

Explanatory: Murder rate per 100k people

Response: Happiness Score (converted to 0 to 100 scale to be in terms of percentage)

1.1 Hypothesized Relationship

Before any analysis is done, we anticipate a negative association between average happiness scores and murder rates. It’s plausible that higher murder rates would coincide with a worse perception of security, resulting in lower happiness rates. There could also be factors such as government instability or even organized crime, which may lower happiness as well as increase murder rates.

1.2 Data Cleaning

In order to find the relationship between murder rates and happiness score for each country per year, we need to merge the total murders data set and population data set. That way, we can get the murder rate per 100K people.

Note

We first have to convert all the values into numbers (i.e. fix cases such as 1.1k to be 1100).

Code 
convert_value <- function(val) {
  val <- as.character(val)
  
  multiplier <- case_when(
    str_detect(val, "k") ~ 1e3,
    str_detect(val, "M") ~ 1e6,
    str_detect(val, "B") ~ 1e9,
    TRUE ~ 1
  )
  
  numeric_value <- as.numeric(str_remove_all(val, "[kMB]"))
  
  return(numeric_value * multiplier)
}

murder_clean <- murder |>
  select(country, `2005`:`2019`) |> 
  pivot_longer(cols = `2005`:`2019`,
               names_to = "year",
               values_to = "murder_count") |> 
  mutate(across(murder_count, ~convert_value(.)))

murder_clean

population_clean <- population |>
  select(country, `2005`:`2019`) |>
  pivot_longer(cols = `2005`:`2019`,
               names_to = "year",
               values_to = "population") |>
  mutate(across(population, ~convert_value(.)))

population_clean

After cleaning the total murders and population data set, we can proceed to merging them to get a data set of the murder rate per 100k people for each country and year. We can then use pivot longer to transform the happiness score data set and merge it with the murder rate per 100k data set to get our final data set.

Code 
murder_pop_merged <- murder_clean |>
  inner_join(population_clean, by = c("country", "year"))

murder_rate_clean <- murder_pop_merged |>
  mutate(murder_rate_per_100k = (murder_count / population) * 100000)

happiness_clean <- happiness |>
  select(country, `2005`:`2019`) |>
  pivot_longer(cols = `2005`:`2019`,
               names_to = "year",
               values_to = "happiness_score") |>
   drop_na(happiness_score)

murder_happiness <- murder_rate_clean |>
  inner_join(happiness_clean, by = c("country", "year"))

murder_happiness |>
  head() |>
  knitr::kable(digits = 4)

The final data set contains 1,820 rows and 6 columns.

The columns are country, year, murder_count, population, murder_rate_per_100k, and happiness_score.

It provides a comprehensive overview of the murder rates and happiness scores across various countries and years. Each entry in the data set corresponds to a unique combination of a country and a year ranging from 2005 to 2019.

2 Linear Regression

In this part, we will use linear regression to model the relationship between our two quantitative variables, murder rate (log) and happiness score. We want to see if our hypothesis from above holds true and whether there is a negative association between the two variables.

2.1 Data Visualization

TODO: edit this, it reads like chatgpt

Here, we will be create two data visualizations that explore the relationship between our two quantitative variables. Through creative visualizations, we aim to gain a deeper understanding of the relationship between the variables and uncover any interesting patterns or trends that may emerge over time and across countries.

2.1.1 Relationship between Murder Rate and Happiness Score Over Time

Code 
animated_plot <- ggplot(murder_happiness,
                        aes(x = log(murder_rate_per_100k),
                            y = happiness_score)) +
  geom_point(color = "steelblue") +
  geom_smooth(method = "lm", color = "black") +
  labs(title = "Relationship Between Murder Rate and Happiness Score (2005-2019)",
       subtitle = "Average Happiness Score",
       x = "Average Murder Rate (per 100k, log scale)",
       y = "",
       caption = "Year: {frame_time}") +
  transition_time(as.integer(year)) +
  enter_fade() +
  exit_fade() +
  theme_bw() +
  theme(plot.caption = element_text(size = 11))

animate(animated_plot, renderer = gifski_renderer())

Throughout the years of 2005 to 2019, there seems to be a consistent negative association between average happiness and murder rate (log).

2.1.2 Relationship Between Murder Rate and Happiness Score

Code 
country_murder_happiness <- murder_happiness |>
  group_by(country) |>
  summarise(avg_murder_rate = mean(murder_rate_per_100k),
            avg_happiness_score = mean(happiness_score))

country_murder_happiness |>
  ggplot(aes(x = log(avg_murder_rate), 
             y = avg_happiness_score)
         ) +
    geom_point(color = "steelblue") +
    geom_smooth(method = "lm", color = "black") +
    labs(title = "Relationship Between Murder Rate and Happiness Score (2005-2019)",
         subtitle = "Average Happiness Score",
         x = "Average Murder Rate (per 100k, log scale)", 
         y = "") +
    theme_bw()

This plot shows the relationship between the average murder rate per 100k for each country and their overall happiness score. As with the previous plot, there is a negative association between the overall average murder rate and happiness.

2.2 Modeling

TODO: Mention conditions being met? maybe in a note block

We will be using linear regression as a statistical method to model the relationship between murder rate and happiness score. Then we will use this to evaluate the model fit.

x (explanatory): average murder rate per 100K people
y (response): average happiness score

Code 
linear_model <- lm(avg_happiness_score~log(avg_murder_rate), country_murder_happiness)
broom::tidy(linear_model) |>
  knitr::kable(digits = 2)
term estimate std.error statistic p.value
(Intercept) 57.77 1.27 45.65 0
log(avg_murder_rate) -2.81 0.71 -3.94 0

\[\begin{equation*} \text{Predicted Happiness Score} = 57.77 - 2.81 \times \ln(\text{Murder Rate}) \end{equation*}\]

The linear regression model suggests that for each each unit increase in the natural logarithm of the average murder rate per 100,000 people, the predicted happiness score decreases by approximately -2.81 points.

2.3 Model Fit

Code 
var_response <- var(country_murder_happiness$avg_happiness_score)
var_fitted <- var(linear_model$fitted.values)
var_resid <- var(linear_model$residuals)
explained_variation <- var_fitted/var_response

table_data <- data.frame(
  Variable = c("Response Variable Variance", "Fitted Values Variance", "Residuals Variance", "Explained Variation (R^2)"),
  Value = c(var_response, var_fitted, var_resid, explained_variation)
)

table_data |>
  knitr::kable(digits = 4)
Variable Value
Response Variable Variance 114.7320
Fitted Values Variance 10.1409
Residuals Variance 104.5911
Explained Variation (R^2) 0.0884

The explained variation is 0.088. This means that murder rate explains about 9 percent of the variability in happiness in our model.

3 Simulation

In this section, we aim to evaluate the performance of our linear regression model by comparing the observed data to simulated data generated using the model’s predictions. By simulating data in this manner, we can assess how well our model captures the underlying relationship between murder rate and happiness.

3.1 Visualizing Simulated Data

The comparison between the observed and simulated data will be visualized through side-by-side plots showcasing the relationships modeled by the linear regression for both data sets.

Code 
set.seed(42)

predictions <- predict(linear_model, country_murder_happiness)
residual_se <- sigma(linear_model)
simulated_y <- predictions + rnorm(n = length(predictions), mean = 0, sd = residual_se)

observed <- ggplot(country_murder_happiness, 
             aes(x = log(avg_murder_rate), 
                 y = avg_happiness_score)
             ) +
  geom_point(color = "steelblue") +
  labs(title = "Observed Data",
       subtitle = "Observed Happiness Score",
       x = "Average Murder Rate", 
       y = "") +
  theme_bw()

# Plot Simulated Data
predicted <- ggplot(country_murder_happiness, 
             aes(x = log(avg_murder_rate), 
                 y = simulated_y)
             ) +
  geom_point(color = "orange3") +
  labs(title = "Simulated Data",
       subtitle = "Simulated Happiness Score",
       x = "Average Murder Rate", 
       y = "") +
  theme_bw()

observed + predicted

TODO: update!

Both the simulated and observed data show clustering at lower murder rates, suggesting a possible non-linear relationship or threshold effect not captured by the linear prediction. However, the simulated data seems to have higher predicted happiness scores at lower murder rates. This discrepancy implies that a linear model provides a simplified representation of the relationship between variables, but it may not accurately capture all the variability and intricacies present in the data set. Therefore, relying solely on a linear model might lead to incomplete or inaccurate conclusions about the underlying phenomena being studied.

3.2 Full Scale Simulation

Code 
set.seed(42)

simulated_ys <- map(1:1000, ~ predictions + rnorm(n = length(predictions), mean = 0, sd = residual_se))

simulated_datasets <- map(simulated_ys, ~ data.frame(avg_murder_rate = log(country_murder_happiness$avg_murder_rate), avg_happiness_score = .x))

# Remove observations with missing values
simulated_datasets <- map(simulated_datasets, na.omit)

# Regress the observed dataset against each simulated dataset and extract R-squared values
r_squared_values <- map_dbl(simulated_datasets, ~ summary(lm(avg_happiness_score ~ avg_murder_rate, data = .x))$r.squared)

# Plot distribution of R-squared values
hist_data <- data.frame(R_squared = r_squared_values)

ggplot(hist_data, aes(x = R_squared)) +
  geom_histogram(fill = "steelblue", color = "black", bins = 30) +
  labs(title = expression("Distribution of"~ R^2 ~"Values"),
       x = expression("Simulated"~ R^2),
       y = "",
       subtitle = "Number of Simulated Models")

TODO: Update!

The histogram shows a right-skewed distribution with a higher frequency on the left side and shorter bars as it moves to the right. We see that the simulated datasets have R-squared values between 0 and 0.075. This suggests that the simulated data shows less variability in happiness scores when comparing with the observed data. It is difficult to ascertain on average how much our simulated data account for the variability in the observed happiness score. But from the graph, one could estimate on average it accounts for <1.25% of the variability in the observed happiness score. The spread of the data also has a narrow spread with values concentrated close to 0, which indicates that the simulated data resembles the observed data.

4 Conclusion

In the end…